There are ways to prevent the "bags of money" from happening. The Foreign Corrupt Practices Act (FCPA) comprehensively prohibits even using the most obscure arrangements to pay bribes. Large institutions hire expensive lawyers to ensure their ongoing compliance with FCPA, because the penalties for failing to prevent your organization from paying bribes are extensive. You can't completely eliminate a practice through law, but you can come close, and FCPA has done more for this problem globally than nearly any other measure enacted by a government.
The Foreign Corrupt Practices Act (FCPA) comprehensively prohibits even using the most obscure arrangements to pay bribes
You can have all the laws you want in words on paper, but if they're not enforced, for all practical purposes, they don't exist.
The people who enforce the FCPA must be understaffed or undermotivated or underfunded because I've worked for several companies that regularly paid bribes as part of doing business.
One example: I worked for a large media company that would send TV crews to cover stories in Mexico on a fairly regular basis. Almost every time the crews tried to return to the United States, the Mexican border personnel would seize their very expensive gear. The only way to get it back was to pay a bribe.
This was so common that everyone was told to just mark it down on their expense reports as "Airport tax." I only found out about it when I started asking why I kept seeing "Airport tax" on expense reports for trips I knew were done in cars.
Your example would be a _very_ far stretch for FCPA.
The law is about bribes for "obtaining or retaining business". It's one thing if you were paying a bribe to say, a local minister to get exclusive access to some sort of scene...
But low-level crooks pretty much sticking you up and you try to buy your stuff back from them under the guise of "government business" is not the kind of thing FCPA is about. It's for concerted attempts to pay off foreign officials to strengthen your business.
Which surely still happen, but not in the manner you're describing. FCPA violations wouldn't be the sort of thing that "everyone" is told about.